Let us mention that the 3.5PN terms in the equations of motion are also known, both for point-particlebinaries[136,137,138,174,148,164]and extended fluid bodies[14,18]; they correspond to 1PN “relative” corrections inthe radiation reaction force. Known also is the contribution of wave tails in the equations of motion, which arises at the 4PNorder and represents a 1.5PN modification of the gravitational radiation damping[27].

Our notation is the following:denotes a multi-index, made of(spatial) indices. Similarly we write forinstance(in practice, we generally do not need to consider the carrier letteror), or. Always understood in expressions such as Equation (25) aresummations over theindicesranging from 1 to 3. The derivative operatoris a short-hand for. The functionis symmetricand trace-free (STF) with respect to theindices composing. This means that for any pair of indices, wehaveand that(see Ref.[210]and Appendices A and B in Ref.[26]for reviewsabout the STF formalism). The STF projection is denoted with a hat, so, or sometimes with carets around theindices,. In particular,is the STF projection of the product of unit vectors; anexpansion into STF tensorsis equivalent to the usual expansion in spherical harmonics.Similarly, we denoteand. Superscripts likeindicatesuccessivetime-derivations.

This assumption is justified because we are ultimately interested in the radiation field at some given finitepost-Newtonian precision like 3PN, and because only a finite number of multipole moments can contribute atany finite order of approximation. With a finite number of multipoles in the linearized metric (26, 27, 28),there is a maximal multipolarityat any post-Minkowskian order, which grows linearly with.

An alternative approach to the problem of radiation reaction, besides the matching procedure, is to work only within apost-Minkowskian iteration scheme (which does not expand the retardations): see, e.g., Ref.[69].

At the 3PN order (taking into account the tails of tails), we find thatdoes not completely cancel out after thereplacement ofby the right-hand side of Equation (100). The reason is that the momentalso depends onatthe 3PN order. Considering also the latter dependence we can check that the 3PN radiative momentis actually free ofthe unphysical constant.

It was shown in Ref.[38]that using one or the other of these derivatives results in some equations of motion that differby a mere coordinate transformation. This result indicates that the distributional derivatives introduced in Ref.[36]constitutemerely some technical tools which are devoid of physical meaning.

Note also that the harmonic-coordinates 3PN equations of motion as they have been obtained in Refs.[37,38]depend, inaddition to, on two arbitrary constantsandparametrizing some logarithmic terms. These constants areclosely related to the constantsandin the partie-finie integral (124); see Ref.[38]for the precisedefinition. However,andare not “physical” in the sense that they can be removed by a coordinatetransformation.

One may wonder why the value ofis a complicated rational fraction whileis so simple. This isbecausewas introduced precisely to measure the amount of ambiguities of certain integrals, while,by contrast,was introduced as an unknown constant entering the relation between the arbitrary scaleson the one hand, andon the other hand, which has a priori nothing to do with ambiguities ofintegrals.

The work[34]provided also some new expressions for the multipole moments of an isolated post-Newtonian source,alternative to those given by Theorem6, in the form of surface integrals extending on the outer part of the source’s nearzone.

When working at the level of the equations of motion (not considering the metric outside the world-lines), the effect ofshifts can be seen as being induced by a coordinate transformation of the bulk metric as in Ref.[38].

Actually, the post-Newtonian series could be only asymptotic (hence divergent), but nevertheless it should give excellentresults provided that the series is truncated near some optimal order of approximation. In this discussion we assume that the3PN order is not too far from that optimum.

When computing the gravitational-wave flux in Ref.[45]we preferred to call the Hadamard-regularization constantsand, in order to distinguish them from the constantsandthat were used in our previous computation of theequations of motion in Ref.[38]. Indeed these regularization constants need not neccessarily be the same when employed indifferent contexts.

Generalizing the flux formula (231) to point masses moving on quasi elliptic orbits dates back to the work of Peters andMathews[178]at Newtonian order. The result was obtained in[217,49]at 1PN order, and then further extended byGopakumar and Iyer[122]up to 2PN order using an explicit quasi-Keplerian representation of the motion[99,197]. Nocomplete result at 3PN order is yet available.